Characterizations of function spaces on the sphere using frames

In this paper we introduce a polynomial frame on the unit sphere S d-1 of R d , for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere S d-1 , such as L p , H p and Besov spaces, can be characterized in terms of the coefficients in the wavelet decompositions, as in the usual Euclidean case R d . We also study a related nonlinear m-term approximation problem on S d-1 . In particular, we prove both a Jackson-type inequality and a Bernstein-type inequality associated to wavelet decompositions, which extend the corresponding results obtained by R. A. DeVore, B. Jawerth and V. Popov ("Compression of wavelet decompositions", Amer. J. Math. 114 (1992), no. 4, 737-785).

[1]  Jochen Göttelmann,et al.  Locally Supported Wavelets on Manifolds with Applications to the 2D Sphere , 1999 .

[2]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[3]  Philippe Jaming,et al.  Harmonic functions on the real hyperbolic ball II Hardy‐Sobolev and Lipschitz spaces , 2004 .

[4]  Willi Freeden,et al.  Constructive Approximation on the Sphere: With Applications to Geomathematics , 1998 .

[5]  Pencho Petrushev,et al.  Localized Tight Frames on Spheres , 2006, SIAM J. Math. Anal..

[6]  Hrushikesh Narhar Mhaskar,et al.  Polynomial Frames: A Fast Tour , 2004 .

[7]  Vladimir N. Temlyakov,et al.  The best m-term approximation and greedy algorithms , 1998, Adv. Comput. Math..

[8]  P. Schröder,et al.  Spherical wavelets: Efficiently representing functions on a sphere , 2000 .

[9]  Philip Crotwell.  Constructive Approximation on the Sphere , 2000 .

[10]  Pierre Vandergheynst,et al.  Wavelets on the n-sphere and related manifolds , 1998 .

[11]  P. Lemarié,et al.  Base d'ondelettes sur les groupes de Lie stratifiés , 1989 .

[12]  Peter Schröder,et al.  Spherical Wavelets: Texture Processing , 1995, Rendering Techniques.

[13]  G. Weiss,et al.  A First Course on Wavelets , 1996 .

[14]  William Lynch,et al.  Compression of functions defined on surfaces of 3D objects , 1997, Proceedings DCC '97. Data Compression Conference.

[15]  R. DeVore,et al.  Nonlinear Wavelet Approximation in the Space C(Rd) , 1992 .

[16]  Guido Weiss,et al.  Maximal estimates for Cesaro and Riesz means on spheres , 1984 .

[17]  S. M. Nikol'skii,et al.  APPROXIMATION OF FUNCTIONS ON THE SPHERE , 1988 .

[18]  A. Bonami,et al.  Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques , 1973 .

[19]  Kh P Rustamov.  ON APPROXIMATION OF FUNCTIONS ON THE SPHERE , 1994 .

[20]  R. DeVore,et al.  Interpolation spaces and non-linear approximation , 1988 .

[21]  Peter Schröder,et al.  Spherical wavelets: efficiently representing functions on the sphere , 1995, SIGGRAPH.

[22]  Tangential characterizations of Hardy and mixed-norm spaces of harmonic functions on the real hyperbolic ball , 2006 .

[23]  M. Skopina,et al.  Polynomial Wavelet-Type Expansions on the Sphere , 2003 .

[24]  Feng Dai,et al.  Approximation of smooth functions on compact two-point homogeneous spaces , 2005, math/0510007.

[25]  F. J. Narcowich,et al.  Nonstationary Wavelets on them-Sphere for Scattered Data , 1996 .

[26]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[27]  W. Dahmen,et al.  Multiresolution analysis and wavelets on S2 and S3 , 1995 .

[28]  R. Jia.  A Bernstein-type inequality associated with wavelet decomposition , 1993 .

[29]  Hrushikesh Narhar Mhaskar,et al.  Polynomial frames on the sphere , 2000, Adv. Comput. Math..

[30]  R. DeVore,et al.  Compression of wavelet decompositions , 1992 .

[31]  Hrushikesh Narhar Mhaskar,et al.  Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature , 2001, Math. Comput..

[32]  B. Jawerth,et al.  A discrete transform and decompositions of distribution spaces , 1990 .

[33]  R. Strichartz.  Multipliers for spherical harmonic expansions , 1972 .